Hey i posted this puzzle a few days ago in another forum, but the stickied thread is completely dead and I don't think anyone even reads it anymore, so I'm going to post it here. Plus I think this is the more proper forum as it is more of a puzzle than a math problem (though it is both).
PUZZLE:
In a very long hallway, there are 1000 doors all initially open.
First, you close every door.
Second, you open every other door.
Next, you toggle the state of every 3rd door (open it if it is closed and close it if it is open),
Next, you toggle every 4th door,
and you continue this process, at the toggling the state of every nth door at the n'th step.
At the end of this process (when n=1000), how many doors are open?
Here is a diagram:
